Well-posedness of the Cauchy problem for multidimensional difference equations in rational cones

The Cauchy Problem for Multidimensional Difference Equations and the Preservation of the Hierarchy of Generating Functions of its Solutions

“…In addition, it can be useful in the study of the stability of the Cauchy problem, since it allows us in some cases to find an explicit form of the fundamental solution of the Cauchy problem as the coefficients of the Laurent series expansion of the function 1/P (z). Note that this played an important role in [2].…”

“…for any functionsφ(x) andg(x). This problem has a unique solution (see [2], Theorem 1) for any fixedφ(x) andg(x). We consider the solutions u(x) of the Cauchy problem (12)- (13) for arguments x lying in the original cone K.…”

“…has a unique solution in an obvious way. For n > 1, the situation is much more complicated and the difficulties are related to the fact that the solution space of the difference equation is infinite-dimensional, and the question of additional conditions that allow to single out unique solutions in this space is non-trivial (see, for example, [1,2,5,8]).…”

“…Conditions on the cone and the point m that ensure solvability, that is, existence and uniqueness, have been studied in papers [2,5,8,13].…”

“…Difference equations are an important tool for investigating combinatorial problems. One way to describe the solution space of a multidimensional difference equation is to specify additional conditions («boundary» or «initial data») for a solution whose fulfillment ensures its existence and uniqueness (the Cauchy problem) (see, for example, [1,2]). The method of generating functions is one of the most advanced theoretical methods of combinatorial analysis and one of the most powerful in applications.…”